Stump the Band
Dear G4BB: I understand that 2 people can be related only by a fraction which has a power of 2 in the bottom [denominator –ed.]…like 2, 4, 8, 16, 32, etc. But given that restriction, is ANY fraction less than 1 possible? …from Matthew Matics Jr. in Space City
92.1 Dear Matt: Off the top of my head, I would say yes. Now granted…”yes” is not a rigorous mathematical proof. I suspect one exists. But maybe not…and if you could show me how it is impossible, say, for 2 people to be 177/512 related, then that would settle it too. But to demonstrate the types of permutations you’d need, I thought I’d sketch out everything from 1/32 to 15/32. 16/32 is of course ½, which is parent/child or brother/sister. Greater than that requires interbreeding…that is, you’d need at least one individual whose parents were related to each other in some way. That would take you from 17/32 up to 31/32…identical twins have the same genetic makeup, so are considered the same person…related by 32/32 or 1.
92.2 I figured this assignment would be tricky but do-able. Then I ran in to a brick wall. Now the cool thing about blogging…as opposed to “dead tree” publishing, like books, magazines, or newspapers…is that if you’re not sure about something, you can still go ahead. No one wants to read a book whose premise is “I have no idea.” But anything goes on the net, and who knows, maybe the answer is out there, and I’d be gratified to hear about it. Or maybe I’ll find the answer myself anyway. But today’s blog is the proverbial “work in progress.”
92.3 So let’s dive right in. 1, 2, 4, and 8 are easily taken care of, in Chart 321. The others will be compounds of these basic relationships…because you and I can be related one way thru our fathers, and another way thru our mothers, and our total CR is the numerical addition of those 2 ways.
Chart 322 outlines what we need, and it looks on the surface to be pretty straightforward.
92.4 The first 3 run pretty much to form, as in Chart 323…father’s side + mother’s side. 7/32 is a little trickier…adding 3 numbers suggests 3 sides? Well, no, 3 lines…and we’ll get to that in a bit.
92.5 Chart 324 gives you the simplest 8’s…notice on the father’s side, X and Y have the same father, thus they are half-siblings, with a CR of 1/4, which is 8/32.
92.6 And Chart 325 extends that idea, converting the mothers’ side into 2 lines…the mothers of X and Y are half-siblings thru their fathers, 1st cousins thru their mothers.
92.7 Chart 326 sums up what we have so far. Find 13, 14, and 15 and we’re done.
92.8 Only now there’s a problem. Let’s analyze how we got 11…we added 8+2+1…translated, X and Y are related by 1/4 or 8/32 on their fathers’ side…X and Y are half-siblings. So far so good. Now X and Y’s mothers are half-siblings thru their fathers, 1/4 or 8/32…and 1st cousins thru their mothers, 1/8 or 4/32. Since X and Y’s mothers’ are half-siblings, X and Y are half-1st cousins…1/16 or 2/32. And since X and Y’s mothers are 1st cousins, X and Y are 2nd cousins…1/32. So that’s 2/32 + 1/32 = 3/32 from their mothers’ side, 8/32 from their fathers’ side, total 11/32. It checks.
92.9 Remember from last week: when 2 people who are related have offspring, the offspring are related to each other by 1/4 the relationship of their parents. And that’s just what happened here…as both half-siblings and 1st cousins, X and Y’s mothers have a total CR of 8/32 + 4/32 = 12/32. And 12/32 divided by 4 gives 3/32…and that’s the CR X and Y get from their mothers’ side.
92.10 Now try applying that to 13 = 8+4+1…the CR that X and Y get from their mothers’ side must be 5/32, so their mothers must be related by 4 times that or 20/32. Well, if the mothers were both siblings, 16/32…and 1st cousins, 4/32…that’s 20/32…divided by 4, gives you 5/32. But can you think of a way, without interbreeding, that the mothers of X and Y can be both siblings and 1st cousins? I can’t. I tried…and I just don’t see it. I’m stuck. Then for 14 = 8+4+2, the mothers must be both siblings and half-siblings…and I don’t even want to think about 15 = 8+4+2+1.
92.11 I stared at Chart 326…how do I extend these patterns? I took it to bed with me…I got up in the morning with it. Nothing budged. One thought I had for the mother’s lines was to “eliminate the middle man”…since going down a generation reduces the CR, why not have X and Y’s mothers be of different generations…mother and daughter instead of sisters. That way, instead of “going down” twice, you only go down once, relative to the older one….you divide by 2, not by 4. But as you can see in Chart 327, it was an interesting idea, but it didn’t help.
92.12 At this point, I wondered if 13, 14, and 15 were possible even with interbreeding.Well, I shouldn’t’ve worried, because they are, as you can see in Chart 328…where the colors of the sexes have been changed, suggesting we’re now in ancient Egypt or someplace.
And in Chart 329, the inner workings are spelled out. For 13, X and Y have married grandparents who are half-siblings…for 14, their married grandparents are full siblings…and for 15, their married great grandparents are full-siblings, as are their married grandparents. And notice what happened with 14…we said in 92.10 that we would need X and Y’s mothers to be both siblings and half-siblings, which is impossible. What they can be is siblings and double 1st cousins…and double 1st cousins have the same CR as half-siblings, 1/4 or 8/32.
92.13 Now one thing you can do when you’re stuck like this, you can “take a running start at it”…by which I mean, go back to simpler cases…see how they work out…and see if that suggests anything. So I retreated from 32nd’s…first back to 8th’s…and that looked kosher up to ½, nothing missing…
92.14 Next, 16th’s…and sure enough, 7/16 is missing…and we can start to sketch out the problem. For 7, we need a CR contribution of 4 from the fathers and 3 from the mothers. Now the fathers’ side is maxed out…X and Y have the same father…their fathers are related to each other by 1, since they’re the same person. And if you can see how 2 people could be more closely related than by being the same person, I’d sure like to hear about it! Using the divide-by-4 rule, X’s father and Y’s father are related by 1…so X and Y are related by 1/4…they are half-brothers thru their father. (Remember, full brothers are half-brothers thru their mothers and their fathers… 1/4 + 1/4 = ½.)
92.15 That leaves 3 we have to get thru X and Y’s mothers…but from the divide-by-4-rule, their mothers would need to be related to each other by 4 times that much…or 12/16…and full sisters are only 8/16. What it’s starting to look like is, you can’t get the CR’s between “3/4 siblings” and full siblings without interbreeding. So for example, with 64th’s you could get 24 for 3/4 siblings, and 32 for full siblings, but you couldn’t get 25 thru 31 without some kind of interbreeding…and if you’d like to check that for me, I’d be super-delighted, you betcha.
92.16 So maybe that’s a small insight, one piece of the puzzle…or maybe I’m completely off base. I feel as if I’m missing something obvious, but I can’t put my finger on it. At the beginning, it seemed reasonable to think that if you could get 16/32 without interbreeding, you wouldn’t need interbreeding to get CR’s less than that…but so far, that intuition appears wrong. Next week, the letters will be really EASY, I can just feel it… 😉 😉 buenas nutcase…
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